Completions of partial matrices

A partial matrix over a field is a matrix whose entries are either elements of the field or independent indeterminates. A completion of a partial matrix is any matrix that results from assigning a field element to each indeterminate. The set of completions of a partial matrix forms an affine subspace of the corresponding space of matrices. This thesis investigates partial matrices whose sets of completions satisfy particular rank properties - specifically partial matrices whose completions all have ranks that are bounded below and partial matrices whose completions all have the same rank. The maximum possible number of indeterminates in such partial matrices are determined and the partial matrices that attain these bounds are fully characterized for all fields. These characterizations utilize a duality between properties of affine spaces of matrices that are related by the trace bilinear form. Precise conditions (based on field order, rank and size) are provided to determine if a partial matrix whose completions all have the same rank r must possess a square partial sub-matrix of order r whose completions are all nonsingular. Finally a characterization of maximal nonsingular partial matrices is provided - a maximal nonsingular partial matrix is a square partial matrix each of whose completions has full rank, with the property that replacement of any constant entry with an indeterminate results in a partial matrix having a singular completion.

Funder

College of Arts, Social Sciences, and Celtic Studies, National University of Ireland, Galway

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Attribution-NonCommercial-NoDerivs 3.0 Ireland

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University of Galway Theses (PhD Theses)